1. Field of the Invention
This invention relates to the field of structural analysis and in particular to the determination of the modal characteristics of a structure.
2. Prior Art
One of the key tools in seeking to prevent unstable modes of vibration in a structure which could cause the destruction of the structure is to experimentally determine such modes and eliminate them through, for example, damping or changes in the configuration of the structure. The initial approach to determine such modes, the passive approach, was to excite the structure with a single shaker, record the response which consisted of whatever superposition of natural modes occurred appropriate to the excitation and analyze the recorded response to separate the overlapping modal motions to estimate characteristics of the individual modes. About twenty-eight years ago a significant breakthrough occurred in the experimental determination of modal characteristics using the passive approach. This breakthrough was provided by Kennedy and Pancu, "Use of Vectors in Vibration Measurement and Analyses," Journal Aeronautical Sciences, Vol. 14, No. 11, Nov. 1947, when they suggested that the comparison of the real versus the imaginary parts of frequency response provided far more discrimination than observation of magnitude. Further refinements followed which eventually resulted in a fairly good agreement of theoretical and measure values of mode shape.
While the passive approach had many advantages, it had a serious disadvantage. When the resonant frequencies of vibration of the structure occurred close to each other, the analytic and/or curve fit techniques became more difficult to apply since the modes were allowed to superimpose in arbitrary combination. The end result was the occurrence of large errors in the determination of the individual modes of the structure. Shortly after the Kennedy and Pancu article, a significant breakthrough in the active approach was made by Lewis and Wrisley, "A System for the Excitation of Pure Natural Modes of Complex Structures," Journal Aeronautical Sciences, Vol. 17, No. 11, Nov. 1950, when they suggested a technique which used multiple shakers and tuned the shakers to induce one mode at a time in the structure. In other words, a spatial distribution of forces were to be applied that matched only one mode so that the resulting response of the structure would be entirely due to a desired mode and no other, thus allowing the characteristics of the mode to be measured without interference from other modes. Further refinements in this technique of multiple shaker model testing followed. Traill-Nash, "On the Excitation of Pure Natural Modes in Aircraft Resonance Testing," Journal Aeronautical Sciences, Vol. 25, No. 12, Dec. 1958, showed how to to calculate the required force distribution from the in-phase sinusoidal response to a set of arbitrary but linearly independent applied forces at the resonant frequency. Asher, "A Method of Normal Mode Excitation Utilizing Admittance Measurements," Proc. National Specialist's Meeting on Dynamics and Aeroelasticity, Inst. Aereonautical Sciences, 1958, Ft. Worth, Tex, relaxed the requirement of knowing in advance the number of degrees of freedom and the exact resonant frequency by iteratively determining them, adding one shaker at a time.
Even with the above refinements, however, it has become increasingly more difficult to conduct a good modal analysis of a structure using the Lewis and Wrisley technique. The Lewis and Wrisley technique requires that a distribution of in-phase (except for polarity) forces be adjusted until an in-phase response is produced, the response then being proportional to the desired mode vector, and that the drive frequency then be shifted until there is an overall 90.degree. phase shift between force and response, that frequency then equalling the desired modal natural resonant frequency. It has been discovered, however, that this technique has certain limitations which intrinsically yields inaccuracies in the determination of the mode vectors and the modal natural resonant frequencies of the structure under analysis when nonproportional damping is present in the structure. It can be shown for an undamped system that the resonant frequencies lie exactly on the j.omega. axis in the frequency domain and that for an undamped, or proportionally damped system the mode vectors are real, that is the vectors are in-phase except for polarity. For a non-proportionally damped system, however, it can be shown that the natural frequencies are not the same as those in the undamped system since the frequencies are in the s or complex frequency plane and the mode vectors are not real but complex and thus are not in-phase. In addition, it can be shown that if the driving force has the right characteristics, the Lewis and Wrisley test criteria will be satisfied for a nonproportionally damped system but the resonant frequencies will be wrong and the response vector will not be the mode vector of the nonproportionally damped system. Finally, it can be shown that with a single frequency, in-phase drive and response, as used in the Lewis and Wrisley system, it is not possible to excite a pure mode of a nonproportionally damped system.
Accordingly, it is a general object of the present invention to provide an improved structural analysis system.
It is another object of the present invention to provide a structural analysis system which can accurately determine the mode vectors and complex resonant frequencies of a structure.
It is yet another object to provide a structural analysis system which accurately determine the mode vectors and complex resonant frequencies of a structure when nonproportonal damping is present in the structure.